LMFDB - Embedded newform 693.4.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 $ x^5 - x^4 - 30x^3 + 21x^2 + 103*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.183399$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.183399 q^{2} -7.96636 q^{4} +17.9271 q^{5} -7.00000 q^{7} +2.92821 q^{8} -3.28780 q^{10} +11.0000 q^{11} -53.2063 q^{13} +1.28379 q^{14} +63.1939 q^{16} +86.7553 q^{17} +55.6681 q^{19} -142.814 q^{20} -2.01738 q^{22} -185.307 q^{23} +196.380 q^{25} +9.75796 q^{26} +55.7646 q^{28} +145.956 q^{29} +132.697 q^{31} -35.0153 q^{32} -15.9108 q^{34} -125.490 q^{35} -129.344 q^{37} -10.2094 q^{38} +52.4942 q^{40} -139.776 q^{41} -87.1438 q^{43} -87.6300 q^{44} +33.9850 q^{46} +576.595 q^{47} +49.0000 q^{49} -36.0158 q^{50} +423.861 q^{52} +92.0572 q^{53} +197.198 q^{55} -20.4975 q^{56} -26.7680 q^{58} -315.667 q^{59} -146.479 q^{61} -24.3365 q^{62} -499.129 q^{64} -953.834 q^{65} +734.048 q^{67} -691.124 q^{68} +23.0146 q^{70} +702.870 q^{71} -361.385 q^{73} +23.7214 q^{74} -443.472 q^{76} -77.0000 q^{77} +1162.09 q^{79} +1132.88 q^{80} +25.6347 q^{82} +1302.46 q^{83} +1555.27 q^{85} +15.9820 q^{86} +32.2103 q^{88} -1383.29 q^{89} +372.444 q^{91} +1476.22 q^{92} -105.747 q^{94} +997.965 q^{95} +212.382 q^{97} -8.98653 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26}+ \cdots + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.183399	−0.0648412	−0.0324206	−	0.999474i	$-0.510322\pi$
$2$	−0.183399	−0.0648412	−0.0324206	+	0.999474i	$0.510322\pi$
$3$	0	0
$3$	0	0
$4$	−7.96636	−0.995796
$5$	17.9271	1.60345	0.801723	−	0.597696i	$-0.203917\pi$
$5$	17.9271	1.60345	0.801723	+	0.597696i	$0.203917\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	2.92821	0.129410
$9$	0	0
$10$	−3.28780	−0.103969
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−53.2063	−1.13514	−0.567568	−	0.823326i	$-0.692116\pi$
$13$	−53.2063	−1.13514	−0.567568	+	0.823326i	$0.692116\pi$
$14$	1.28379	0.0245077
$15$	0	0
$16$	63.1939	0.987405
$17$	86.7553	1.23772	0.618860	−	0.785501i	$-0.287595\pi$
$17$	86.7553	1.23772	0.618860	+	0.785501i	$0.287595\pi$
$18$	0	0
$19$	55.6681	0.672165	0.336082	−	0.941833i	$-0.390898\pi$
$19$	55.6681	0.672165	0.336082	+	0.941833i	$0.390898\pi$
$20$	−142.814	−1.59670
$21$	0	0
$22$	−2.01738	−0.0195504
$23$	−185.307	−1.67996	−0.839981	−	0.542616i	$-0.817434\pi$
$23$	−185.307	−1.67996	−0.839981	+	0.542616i	$0.817434\pi$
$24$	0	0
$25$	196.380	1.57104
$26$	9.75796	0.0736036
$27$	0	0
$28$	55.7646	0.376375
$29$	145.956	0.934595	0.467298	−	0.884100i	$-0.345228\pi$
$29$	145.956	0.934595	0.467298	+	0.884100i	$0.345228\pi$
$30$	0	0
$31$	132.697	0.768810	0.384405	−	0.923165i	$-0.374407\pi$
$31$	132.697	0.768810	0.384405	+	0.923165i	$0.374407\pi$
$32$	−35.0153	−0.193434
$33$	0	0
$34$	−15.9108	−0.0802553
$35$	−125.490	−0.606046
$36$	0	0
$37$	−129.344	−0.574701	−0.287351	−	0.957825i	$-0.592774\pi$
$37$	−129.344	−0.574701	−0.287351	+	0.957825i	$0.592774\pi$
$38$	−10.2094	−0.0435840
$39$	0	0
$40$	52.4942	0.207502
$41$	−139.776	−0.532423	−0.266212	−	0.963915i	$-0.585772\pi$
$41$	−139.776	−0.532423	−0.266212	+	0.963915i	$0.585772\pi$
$42$	0	0
$43$	−87.1438	−0.309053	−0.154527	−	0.987989i	$-0.549385\pi$
$43$	−87.1438	−0.309053	−0.154527	+	0.987989i	$0.549385\pi$
$44$	−87.6300	−0.300244
$45$	0	0
$46$	33.9850	0.108931
$47$	576.595	1.78947	0.894734	−	0.446599i	$-0.147365\pi$
$47$	576.595	1.78947	0.894734	+	0.446599i	$0.147365\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−36.0158	−0.101868
$51$	0	0
$52$	423.861	1.13036
$53$	92.0572	0.238585	0.119293	−	0.992859i	$-0.461937\pi$
$53$	92.0572	0.238585	0.119293	+	0.992859i	$0.461937\pi$
$54$	0	0
$55$	197.198	0.483457
$56$	−20.4975	−0.0489123
$57$	0	0
$58$	−26.7680	−0.0606003
$59$	−315.667	−0.696547	−0.348274	−	0.937393i	$-0.613232\pi$
$59$	−315.667	−0.696547	−0.348274	+	0.937393i	$0.613232\pi$
$60$	0	0
$61$	−146.479	−0.307455	−0.153727	−	0.988113i	$-0.549128\pi$
$61$	−146.479	−0.307455	−0.153727	+	0.988113i	$0.549128\pi$
$62$	−24.3365	−0.0498505
$63$	0	0
$64$	−499.129	−0.974862
$65$	−953.834	−1.82013
$66$	0	0
$67$	734.048	1.33848	0.669241	−	0.743046i	$-0.266620\pi$
$67$	734.048	1.33848	0.669241	+	0.743046i	$0.266620\pi$
$68$	−691.124	−1.23252
$69$	0	0
$70$	23.0146	0.0392967
$71$	702.870	1.17486	0.587432	−	0.809274i	$-0.300139\pi$
$71$	702.870	1.17486	0.587432	+	0.809274i	$0.300139\pi$
$72$	0	0
$73$	−361.385	−0.579410	−0.289705	−	0.957116i	$-0.593557\pi$
$73$	−361.385	−0.579410	−0.289705	+	0.957116i	$0.593557\pi$
$74$	23.7214	0.0372643
$75$	0	0
$76$	−443.472	−0.669339
$77$	−77.0000	−0.113961
$78$	0	0
$79$	1162.09	1.65501	0.827503	−	0.561461i	$-0.189761\pi$
$79$	1162.09	1.65501	0.827503	+	0.561461i	$0.189761\pi$
$80$	1132.88	1.58325
$81$	0	0
$82$	25.6347	0.0345230
$83$	1302.46	1.72245	0.861223	−	0.508227i	$-0.169699\pi$
$83$	1302.46	1.72245	0.861223	+	0.508227i	$0.169699\pi$
$84$	0	0
$85$	1555.27	1.98462
$86$	15.9820	0.0200394
$87$	0	0
$88$	32.2103	0.0390185
$89$	−1383.29	−1.64751	−0.823757	−	0.566942i	$-0.808126\pi$
$89$	−1383.29	−1.64751	−0.823757	+	0.566942i	$0.808126\pi$
$90$	0	0
$91$	372.444	0.429041
$92$	1476.22	1.67290
$93$	0	0
$94$	−105.747	−0.116031
$95$	997.965	1.07778
$96$	0	0
$97$	212.382	0.222310	0.111155	−	0.993803i	$-0.464545\pi$
$97$	212.382	0.222310	0.111155	+	0.993803i	$0.464545\pi$
$98$	−8.98653	−0.00926303
$99$	0	0
$100$	−1564.43	−1.56443
$101$	1326.57	1.30692	0.653459	−	0.756962i	$-0.273317\pi$
$101$	1326.57	1.30692	0.653459	+	0.756962i	$0.273317\pi$
$102$	0	0
$103$	1360.38	1.30138	0.650688	−	0.759345i	$-0.274480\pi$
$103$	1360.38	1.30138	0.650688	+	0.759345i	$0.274480\pi$
$104$	−155.799	−0.146898
$105$	0	0
$106$	−16.8832	−0.0154702
$107$	667.188	0.602799	0.301400	−	0.953498i	$-0.402546\pi$
$107$	667.188	0.602799	0.301400	+	0.953498i	$0.402546\pi$
$108$	0	0
$109$	1208.14	1.06164	0.530822	−	0.847484i	$-0.321883\pi$
$109$	1208.14	1.06164	0.530822	+	0.847484i	$0.321883\pi$
$110$	−36.1658	−0.0313479
$111$	0	0
$112$	−442.357	−0.373204
$113$	889.537	0.740536	0.370268	−	0.928925i	$-0.379266\pi$
$113$	889.537	0.740536	0.370268	+	0.928925i	$0.379266\pi$
$114$	0	0
$115$	−3322.01	−2.69373
$116$	−1162.74	−0.930666
$117$	0	0
$118$	57.8928	0.0451650
$119$	−607.287	−0.467814
$120$	0	0
$121$	121.000	0.0909091
$122$	26.8641	0.0199357
$123$	0	0
$124$	−1057.11	−0.765577
$125$	1279.63	0.915631
$126$	0	0
$127$	2474.54	1.72898	0.864489	−	0.502652i	$-0.167642\pi$
$127$	2474.54	1.72898	0.864489	+	0.502652i	$0.167642\pi$
$128$	371.662	0.256645
$129$	0	0
$130$	174.932	0.118019
$131$	−307.112	−0.204829	−0.102414	−	0.994742i	$-0.532657\pi$
$131$	−307.112	−0.204829	−0.102414	+	0.994742i	$0.532657\pi$
$132$	0	0
$133$	−389.676	−0.254054
$134$	−134.623	−0.0867887
$135$	0	0
$136$	254.038	0.160173
$137$	−301.007	−0.187714	−0.0938569	−	0.995586i	$-0.529920\pi$
$137$	−301.007	−0.187714	−0.0938569	+	0.995586i	$0.529920\pi$
$138$	0	0
$139$	951.210	0.580436	0.290218	−	0.956961i	$-0.406272\pi$
$139$	951.210	0.580436	0.290218	+	0.956961i	$0.406272\pi$
$140$	999.695	0.603498
$141$	0	0
$142$	−128.905	−0.0761796
$143$	−585.270	−0.342257
$144$	0	0
$145$	2616.56	1.49857
$146$	66.2775	0.0375696
$147$	0	0
$148$	1030.40	0.572285
$149$	672.616	0.369818	0.184909	−	0.982756i	$-0.440801\pi$
$149$	672.616	0.369818	0.184909	+	0.982756i	$0.440801\pi$
$150$	0	0
$151$	−2367.48	−1.27591	−0.637955	−	0.770074i	$-0.720219\pi$
$151$	−2367.48	−1.27591	−0.637955	+	0.770074i	$0.720219\pi$
$152$	163.008	0.0869847
$153$	0	0
$154$	14.1217	0.00738934
$155$	2378.87	1.23274
$156$	0	0
$157$	1375.49	0.699211	0.349605	−	0.936897i	$-0.386316\pi$
$157$	1375.49	0.699211	0.349605	+	0.936897i	$0.386316\pi$
$158$	−213.126	−0.107313
$159$	0	0
$160$	−627.723	−0.310161
$161$	1297.15	0.634966
$162$	0	0
$163$	2441.46	1.17319	0.586594	−	0.809881i	$-0.300469\pi$
$163$	2441.46	1.17319	0.586594	+	0.809881i	$0.300469\pi$
$164$	1113.51	0.530185
$165$	0	0
$166$	−238.868	−0.111685
$167$	−13.6322	−0.00631670	−0.00315835	−	0.999995i	$-0.501005\pi$
$167$	−13.6322	−0.00631670	−0.00315835	+	0.999995i	$0.501005\pi$
$168$	0	0
$169$	633.912	0.288535
$170$	−285.234	−0.128685
$171$	0	0
$172$	694.219	0.307754
$173$	−3685.65	−1.61974	−0.809870	−	0.586609i	$-0.800463\pi$
$173$	−3685.65	−1.61974	−0.809870	+	0.586609i	$0.800463\pi$
$174$	0	0
$175$	−1374.66	−0.593797
$176$	695.133	0.297714
$177$	0	0
$178$	253.694	0.106827
$179$	−2042.61	−0.852915	−0.426457	−	0.904508i	$-0.640239\pi$
$179$	−2042.61	−0.852915	−0.426457	+	0.904508i	$0.640239\pi$
$180$	0	0
$181$	−4414.29	−1.81277	−0.906386	−	0.422450i	$-0.861170\pi$
$181$	−4414.29	−1.81277	−0.906386	+	0.422450i	$0.861170\pi$
$182$	−68.3058	−0.0278196
$183$	0	0
$184$	−542.617	−0.217403
$185$	−2318.75	−0.921502
$186$	0	0
$187$	954.308	0.373187
$188$	−4593.36	−1.78194
$189$	0	0
$190$	−183.025	−0.0698845
$191$	−3300.03	−1.25017	−0.625084	−	0.780558i	$-0.714935\pi$
$191$	−3300.03	−1.25017	−0.625084	+	0.780558i	$0.714935\pi$
$192$	0	0
$193$	3368.36	1.25627	0.628135	−	0.778105i	$-0.283819\pi$
$193$	3368.36	1.25627	0.628135	+	0.778105i	$0.283819\pi$
$194$	−38.9505	−0.0144149
$195$	0	0
$196$	−390.352	−0.142257
$197$	1367.51	0.494574	0.247287	−	0.968942i	$-0.420461\pi$
$197$	1367.51	0.494574	0.247287	+	0.968942i	$0.420461\pi$
$198$	0	0
$199$	−4612.02	−1.64290	−0.821450	−	0.570280i	$-0.806835\pi$
$199$	−4612.02	−1.64290	−0.821450	+	0.570280i	$0.806835\pi$
$200$	575.041	0.203308
$201$	0	0
$202$	−243.291	−0.0847421
$203$	−1021.69	−0.353244
$204$	0	0
$205$	−2505.78	−0.853712
$206$	−249.491	−0.0843828
$207$	0	0
$208$	−3362.31	−1.12084
$209$	612.349	0.202665
$210$	0	0
$211$	4838.75	1.57874	0.789368	−	0.613921i	$-0.210409\pi$
$211$	4838.75	1.57874	0.789368	+	0.613921i	$0.210409\pi$
$212$	−733.361	−0.237582
$213$	0	0
$214$	−122.361	−0.0390862
$215$	−1562.23	−0.495551
$216$	0	0
$217$	−928.879	−0.290583
$218$	−221.572	−0.0688382
$219$	0	0
$220$	−1570.95	−0.481425
$221$	−4615.93	−1.40498
$222$	0	0
$223$	1794.33	0.538822	0.269411	−	0.963025i	$-0.413171\pi$
$223$	1794.33	0.538822	0.269411	+	0.963025i	$0.413171\pi$
$224$	245.107	0.0731113
$225$	0	0
$226$	−163.140	−0.0480173
$227$	1297.05	0.379243	0.189622	−	0.981857i	$-0.439274\pi$
$227$	1297.05	0.379243	0.189622	+	0.981857i	$0.439274\pi$
$228$	0	0
$229$	2630.13	0.758968	0.379484	−	0.925198i	$-0.376102\pi$
$229$	2630.13	0.758968	0.379484	+	0.925198i	$0.376102\pi$
$230$	609.251	0.174665
$231$	0	0
$232$	427.388	0.120946
$233$	−3707.36	−1.04239	−0.521196	−	0.853437i	$-0.674514\pi$
$233$	−3707.36	−1.04239	−0.521196	+	0.853437i	$0.674514\pi$
$234$	0	0
$235$	10336.7	2.86932
$236$	2514.72	0.693619
$237$	0	0
$238$	111.376	0.0303336
$239$	−2844.60	−0.769882	−0.384941	−	0.922941i	$-0.625778\pi$
$239$	−2844.60	−0.769882	−0.384941	+	0.922941i	$0.625778\pi$
$240$	0	0
$241$	1778.70	0.475419	0.237710	−	0.971336i	$-0.423603\pi$
$241$	1778.70	0.475419	0.237710	+	0.971336i	$0.423603\pi$
$242$	−22.1912	−0.00589465
$243$	0	0
$244$	1166.91	0.306162
$245$	878.427	0.229064
$246$	0	0
$247$	−2961.89	−0.762999
$248$	388.565	0.0994915
$249$	0	0
$250$	−234.683	−0.0593706
$251$	−254.840	−0.0640851	−0.0320426	−	0.999487i	$-0.510201\pi$
$251$	−254.840	−0.0640851	−0.0320426	+	0.999487i	$0.510201\pi$
$252$	0	0
$253$	−2038.37	−0.506528
$254$	−453.828	−0.112109
$255$	0	0
$256$	3924.87	0.958221
$257$	1617.63	0.392626	0.196313	−	0.980541i	$-0.437103\pi$
$257$	1617.63	0.392626	0.196313	+	0.980541i	$0.437103\pi$
$258$	0	0
$259$	905.405	0.217217
$260$	7598.59	1.81248
$261$	0	0
$262$	56.3240	0.0132813
$263$	2738.61	0.642090	0.321045	−	0.947064i	$-0.395966\pi$
$263$	2738.61	0.642090	0.321045	+	0.947064i	$0.395966\pi$
$264$	0	0
$265$	1650.32	0.382559
$266$	71.4661	0.0164732
$267$	0	0
$268$	−5847.70	−1.33285
$269$	856.853	0.194213	0.0971064	−	0.995274i	$-0.469041\pi$
$269$	856.853	0.194213	0.0971064	+	0.995274i	$0.469041\pi$
$270$	0	0
$271$	−6081.47	−1.36318	−0.681592	−	0.731732i	$-0.738712\pi$
$271$	−6081.47	−1.36318	−0.681592	+	0.731732i	$0.738712\pi$
$272$	5482.40	1.22213
$273$	0	0
$274$	55.2043	0.0121716
$275$	2160.18	0.473686
$276$	0	0
$277$	−623.366	−0.135215	−0.0676073	−	0.997712i	$-0.521536\pi$
$277$	−623.366	−0.135215	−0.0676073	+	0.997712i	$0.521536\pi$
$278$	−174.451	−0.0376361
$279$	0	0
$280$	−367.460	−0.0784282
$281$	−1273.56	−0.270370	−0.135185	−	0.990820i	$-0.543163\pi$
$281$	−1273.56	−0.270370	−0.135185	+	0.990820i	$0.543163\pi$
$282$	0	0
$283$	−8224.31	−1.72751	−0.863753	−	0.503915i	$-0.831893\pi$
$283$	−8224.31	−1.72751	−0.863753	+	0.503915i	$0.831893\pi$
$284$	−5599.32	−1.16992
$285$	0	0
$286$	107.338	0.0221923
$287$	978.432	0.201237
$288$	0	0
$289$	2613.48	0.531952
$290$	−479.873	−0.0971693
$291$	0	0
$292$	2878.93	0.576974
$293$	−3203.94	−0.638826	−0.319413	−	0.947616i	$-0.603486\pi$
$293$	−3203.94	−0.638826	−0.319413	+	0.947616i	$0.603486\pi$
$294$	0	0
$295$	−5658.98	−1.11688
$296$	−378.745	−0.0743719
$297$	0	0
$298$	−123.357	−0.0239794
$299$	9859.49	1.90699
$300$	0	0
$301$	610.006	0.116811
$302$	434.192	0.0827315
$303$	0	0
$304$	3517.88	0.663698
$305$	−2625.94	−0.492987
$306$	0	0
$307$	−7457.11	−1.38632	−0.693159	−	0.720784i	$-0.743782\pi$
$307$	−7457.11	−1.38632	−0.693159	+	0.720784i	$0.743782\pi$
$308$	613.410	0.113481
$309$	0	0
$310$	−436.281	−0.0799326
$311$	2732.06	0.498137	0.249069	−	0.968486i	$-0.419876\pi$
$311$	2732.06	0.498137	0.249069	+	0.968486i	$0.419876\pi$
$312$	0	0
$313$	6263.36	1.13107	0.565537	−	0.824723i	$-0.308669\pi$
$313$	6263.36	1.13107	0.565537	+	0.824723i	$0.308669\pi$
$314$	−252.263	−0.0453377
$315$	0	0
$316$	−9257.64	−1.64805
$317$	−2732.41	−0.484125	−0.242062	−	0.970261i	$-0.577824\pi$
$317$	−2732.41	−0.484125	−0.242062	+	0.970261i	$0.577824\pi$
$318$	0	0
$319$	1605.51	0.281791
$320$	−8947.93	−1.56314
$321$	0	0
$322$	−237.895	−0.0411719
$323$	4829.50	0.831952
$324$	0	0
$325$	−10448.7	−1.78334
$326$	−447.759	−0.0760709
$327$	0	0
$328$	−409.294	−0.0689008
$329$	−4036.16	−0.676355
$330$	0	0
$331$	−2578.92	−0.428249	−0.214124	−	0.976806i	$-0.568690\pi$
$331$	−2578.92	−0.428249	−0.214124	+	0.976806i	$0.568690\pi$
$332$	−10375.8	−1.71520
$333$	0	0
$334$	2.50012	0.000409582 0
$335$	13159.3	2.14618
$336$	0	0
$337$	−9297.79	−1.50292	−0.751458	−	0.659780i	$-0.770649\pi$
$337$	−9297.79	−1.50292	−0.751458	+	0.659780i	$0.770649\pi$
$338$	−116.259	−0.0187090
$339$	0	0
$340$	−12389.8	−1.97627
$341$	1459.67	0.231805
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−255.175	−0.0399945
$345$	0	0
$346$	675.944	0.105026
$347$	6167.17	0.954096	0.477048	−	0.878877i	$-0.341707\pi$
$347$	6167.17	0.954096	0.477048	+	0.878877i	$0.341707\pi$
$348$	0	0
$349$	10711.8	1.64294	0.821472	−	0.570249i	$-0.193153\pi$
$349$	10711.8	1.64294	0.821472	+	0.570249i	$0.193153\pi$
$350$	252.111	0.0385025
$351$	0	0
$352$	−385.169	−0.0583226
$353$	−5236.26	−0.789512	−0.394756	−	0.918786i	$-0.629171\pi$
$353$	−5236.26	−0.789512	−0.394756	+	0.918786i	$0.629171\pi$
$354$	0	0
$355$	12600.4	1.88383
$356$	11019.8	1.64059
$357$	0	0
$358$	374.612	0.0553040
$359$	7822.89	1.15007	0.575037	−	0.818127i	$-0.304988\pi$
$359$	7822.89	1.15007	0.575037	+	0.818127i	$0.304988\pi$
$360$	0	0
$361$	−3760.07	−0.548195
$362$	809.575	0.117542
$363$	0	0
$364$	−2967.03	−0.427238
$365$	−6478.58	−0.929052
$366$	0	0
$367$	−6533.92	−0.929340	−0.464670	−	0.885484i	$-0.653827\pi$
$367$	−6533.92	−0.929340	−0.464670	+	0.885484i	$0.653827\pi$
$368$	−11710.3	−1.65880
$369$	0	0
$370$	425.256	0.0597513
$371$	−644.400	−0.0901768
$372$	0	0
$373$	−7348.39	−1.02007	−0.510034	−	0.860154i	$-0.670367\pi$
$373$	−7348.39	−1.02007	−0.510034	+	0.860154i	$0.670367\pi$
$374$	−175.019	−0.0241979
$375$	0	0
$376$	1688.39	0.231575
$377$	−7765.76	−1.06089
$378$	0	0
$379$	−2549.40	−0.345525	−0.172762	−	0.984964i	$-0.555269\pi$
$379$	−2549.40	−0.345525	−0.172762	+	0.984964i	$0.555269\pi$
$380$	−7950.16	−1.07325
$381$	0	0
$382$	605.221	0.0810623
$383$	3017.93	0.402634	0.201317	−	0.979526i	$-0.435478\pi$
$383$	3017.93	0.402634	0.201317	+	0.979526i	$0.435478\pi$
$384$	0	0
$385$	−1380.38	−0.182730
$386$	−617.753	−0.0814580
$387$	0	0
$388$	−1691.91	−0.221376
$389$	−7719.08	−1.00610	−0.503050	−	0.864257i	$-0.667789\pi$
$389$	−7719.08	−1.00610	−0.503050	+	0.864257i	$0.667789\pi$
$390$	0	0
$391$	−16076.3	−2.07932
$392$	143.482	0.0184871
$393$	0	0
$394$	−250.799	−0.0320687
$395$	20832.9	2.65371
$396$	0	0
$397$	−7765.04	−0.981652	−0.490826	−	0.871258i	$-0.663305\pi$
$397$	−7765.04	−0.981652	−0.490826	+	0.871258i	$0.663305\pi$
$398$	845.837	0.106528
$399$	0	0
$400$	12410.0	1.55125
$401$	−11406.1	−1.42044	−0.710219	−	0.703981i	$-0.751404\pi$
$401$	−11406.1	−1.42044	−0.710219	+	0.703981i	$0.751404\pi$
$402$	0	0
$403$	−7060.32	−0.872704
$404$	−10567.9	−1.30142
$405$	0	0
$406$	187.376	0.0229048
$407$	−1422.78	−0.173279
$408$	0	0
$409$	8201.52	0.991538	0.495769	−	0.868454i	$-0.334886\pi$
$409$	8201.52	0.991538	0.495769	+	0.868454i	$0.334886\pi$
$410$	459.556	0.0553557
$411$	0	0
$412$	−10837.2	−1.29590
$413$	2209.67	0.263270
$414$	0	0
$415$	23349.2	2.76185
$416$	1863.04	0.219574
$417$	0	0
$418$	−112.304	−0.0131411
$419$	−15833.2	−1.84607	−0.923036	−	0.384715i	$-0.874300\pi$
$419$	−15833.2	−1.84607	−0.923036	+	0.384715i	$0.874300\pi$
$420$	0	0
$421$	−11030.9	−1.27699	−0.638497	−	0.769624i	$-0.720443\pi$
$421$	−11030.9	−1.27699	−0.638497	+	0.769624i	$0.720443\pi$
$422$	−887.420	−0.102367
$423$	0	0
$424$	269.563	0.0308753
$425$	17037.0	1.94451
$426$	0	0
$427$	1025.36	0.116207
$428$	−5315.06	−0.600265
$429$	0	0
$430$	286.511	0.0321321
$431$	−13511.0	−1.50998	−0.754990	−	0.655737i	$-0.772358\pi$
$431$	−13511.0	−1.50998	−0.754990	+	0.655737i	$0.772358\pi$
$432$	0	0
$433$	9540.73	1.05889	0.529443	−	0.848345i	$-0.322401\pi$
$433$	9540.73	1.05889	0.529443	+	0.848345i	$0.322401\pi$
$434$	170.355	0.0188417
$435$	0	0
$436$	−9624.51	−1.05718
$437$	−10315.7	−1.12921
$438$	0	0
$439$	16487.1	1.79246	0.896228	−	0.443593i	$-0.146297\pi$
$439$	16487.1	1.79246	0.896228	+	0.443593i	$0.146297\pi$
$440$	577.436	0.0625641
$441$	0	0
$442$	846.555	0.0911007
$443$	15708.7	1.68475	0.842376	−	0.538891i	$-0.181156\pi$
$443$	15708.7	1.68475	0.842376	+	0.538891i	$0.181156\pi$
$444$	0	0
$445$	−24798.4	−2.64170
$446$	−329.078	−0.0349379
$447$	0	0
$448$	3493.91	0.368463
$449$	−9850.65	−1.03537	−0.517685	−	0.855572i	$-0.673206\pi$
$449$	−9850.65	−1.03537	−0.517685	+	0.855572i	$0.673206\pi$
$450$	0	0
$451$	−1537.54	−0.160532
$452$	−7086.38	−0.737423
$453$	0	0
$454$	−237.877	−0.0245906
$455$	6676.83	0.687945
$456$	0	0
$457$	5247.90	0.537170	0.268585	−	0.963256i	$-0.413444\pi$
$457$	5247.90	0.537170	0.268585	+	0.963256i	$0.413444\pi$
$458$	−482.362	−0.0492124
$459$	0	0
$460$	26464.3	2.68240
$461$	−5258.64	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$461$	−5258.64	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$462$	0	0
$463$	8815.88	0.884900	0.442450	−	0.896793i	$-0.354109\pi$
$463$	8815.88	0.884900	0.442450	+	0.896793i	$0.354109\pi$
$464$	9223.50	0.922824
$465$	0	0
$466$	679.925	0.0675899
$467$	−2599.62	−0.257594	−0.128797	−	0.991671i	$-0.541111\pi$
$467$	−2599.62	−0.257594	−0.128797	+	0.991671i	$0.541111\pi$
$468$	0	0
$469$	−5138.34	−0.505898
$470$	−1895.73	−0.186050
$471$	0	0
$472$	−924.338	−0.0901400
$473$	−958.581	−0.0931831
$474$	0	0
$475$	10932.1	1.05600
$476$	4837.87	0.465847
$477$	0	0
$478$	521.695	0.0499200
$479$	6316.67	0.602539	0.301269	−	0.953539i	$-0.402590\pi$
$479$	6316.67	0.602539	0.301269	+	0.953539i	$0.402590\pi$
$480$	0	0
$481$	6881.89	0.652364
$482$	−326.211	−0.0308268
$483$	0	0
$484$	−963.930	−0.0905269
$485$	3807.39	0.356463
$486$	0	0
$487$	−6664.47	−0.620115	−0.310058	−	0.950718i	$-0.600348\pi$
$487$	−6664.47	−0.620115	−0.310058	+	0.950718i	$0.600348\pi$
$488$	−428.922	−0.0397877
$489$	0	0
$490$	−161.102	−0.0148528
$491$	−6411.20	−0.589274	−0.294637	−	0.955609i	$-0.595199\pi$
$491$	−6411.20	−0.589274	−0.294637	+	0.955609i	$0.595199\pi$
$492$	0	0
$493$	12662.4	1.15677
$494$	543.207	0.0494738
$495$	0	0
$496$	8385.64	0.759126
$497$	−4920.09	−0.444057
$498$	0	0
$499$	−6855.85	−0.615050	−0.307525	−	0.951540i	$-0.599501\pi$
$499$	−6855.85	−0.615050	−0.307525	+	0.951540i	$0.599501\pi$
$500$	−10194.0	−0.911781
$501$	0	0
$502$	46.7373	0.00415536
$503$	12991.9	1.15165	0.575827	−	0.817572i	$-0.304680\pi$
$503$	12991.9	1.15165	0.575827	+	0.817572i	$0.304680\pi$
$504$	0	0
$505$	23781.5	2.09557
$506$	373.835	0.0328439
$507$	0	0
$508$	−19713.1	−1.72171
$509$	8619.95	0.750633	0.375317	−	0.926897i	$-0.377534\pi$
$509$	8619.95	0.750633	0.375317	+	0.926897i	$0.377534\pi$
$510$	0	0
$511$	2529.70	0.218996
$512$	−3693.12	−0.318778
$513$	0	0
$514$	−296.671	−0.0254583
$515$	24387.5	2.08669
$516$	0	0
$517$	6342.54	0.539545
$518$	−166.050	−0.0140846
$519$	0	0
$520$	−2793.02	−0.235543
$521$	−79.1785	−0.00665810	−0.00332905	−	0.999994i	$-0.501060\pi$
$521$	−79.1785	−0.00665810	−0.00332905	+	0.999994i	$0.501060\pi$
$522$	0	0
$523$	11304.4	0.945141	0.472571	−	0.881293i	$-0.343326\pi$
$523$	11304.4	0.945141	0.472571	+	0.881293i	$0.343326\pi$
$524$	2446.57	0.203967
$525$	0	0
$526$	−502.256	−0.0416339
$527$	11512.2	0.951571
$528$	0	0
$529$	22171.6	1.82227
$530$	−302.666	−0.0248056
$531$	0	0
$532$	3104.30	0.252986
$533$	7436.97	0.604373
$534$	0	0
$535$	11960.7	0.966556
$536$	2149.45	0.173213
$537$	0	0
$538$	−157.146	−0.0125930
$539$	539.000	0.0430730
$540$	0	0
$541$	−5184.07	−0.411979	−0.205989	−	0.978554i	$-0.566041\pi$
$541$	−5184.07	−0.411979	−0.205989	+	0.978554i	$0.566041\pi$
$542$	1115.33	0.0883905
$543$	0	0
$544$	−3037.77	−0.239418
$545$	21658.5	1.70229
$546$	0	0
$547$	18111.1	1.41568	0.707839	−	0.706374i	$-0.249670\pi$
$547$	18111.1	1.41568	0.707839	+	0.706374i	$0.249670\pi$
$548$	2397.93	0.186924
$549$	0	0
$550$	−396.174	−0.0307144
$551$	8125.06	0.628202
$552$	0	0
$553$	−8134.64	−0.625533
$554$	114.324	0.00876747
$555$	0	0
$556$	−7577.68	−0.577995
$557$	20608.7	1.56772	0.783859	−	0.620939i	$-0.213249\pi$
$557$	20608.7	1.56772	0.783859	+	0.620939i	$0.213249\pi$
$558$	0	0
$559$	4636.60	0.350818
$560$	−7930.17	−0.598412
$561$	0	0
$562$	233.569	0.0175311
$563$	−9840.79	−0.736660	−0.368330	−	0.929695i	$-0.620070\pi$
$563$	−9840.79	−0.736660	−0.368330	+	0.929695i	$0.620070\pi$
$564$	0	0
$565$	15946.8	1.18741
$566$	1508.33	0.112014
$567$	0	0
$568$	2058.15	0.152039
$569$	−12936.3	−0.953104	−0.476552	−	0.879146i	$-0.658114\pi$
$569$	−12936.3	−0.953104	−0.476552	+	0.879146i	$0.658114\pi$
$570$	0	0
$571$	20105.5	1.47353	0.736767	−	0.676146i	$-0.236351\pi$
$571$	20105.5	1.47353	0.736767	+	0.676146i	$0.236351\pi$
$572$	4662.47	0.340818
$573$	0	0
$574$	−179.443	−0.0130485
$575$	−36390.5	−2.63929
$576$	0	0
$577$	−4071.91	−0.293788	−0.146894	−	0.989152i	$-0.546928\pi$
$577$	−4071.91	−0.293788	−0.146894	+	0.989152i	$0.546928\pi$
$578$	−479.308	−0.0344924
$579$	0	0
$580$	−20844.4	−1.49227
$581$	−9117.19	−0.651023
$582$	0	0
$583$	1012.63	0.0719362
$584$	−1058.21	−0.0749813
$585$	0	0
$586$	587.598	0.0414223
$587$	−4603.50	−0.323691	−0.161846	−	0.986816i	$-0.551745\pi$
$587$	−4603.50	−0.323691	−0.161846	+	0.986816i	$0.551745\pi$
$588$	0	0
$589$	7386.99	0.516767
$590$	1037.85	0.0724196
$591$	0	0
$592$	−8173.72	−0.567462
$593$	20033.1	1.38728	0.693642	−	0.720320i	$-0.256005\pi$
$593$	20033.1	1.38728	0.693642	+	0.720320i	$0.256005\pi$
$594$	0	0
$595$	−10886.9	−0.750115
$596$	−5358.30	−0.368263
$597$	0	0
$598$	−1808.22	−0.123651
$599$	14558.0	0.993028	0.496514	−	0.868029i	$-0.334613\pi$
$599$	14558.0	0.993028	0.496514	+	0.868029i	$0.334613\pi$
$600$	0	0
$601$	20804.2	1.41201	0.706007	−	0.708205i	$-0.250495\pi$
$601$	20804.2	1.41201	0.706007	+	0.708205i	$0.250495\pi$
$602$	−111.874	−0.00757418
$603$	0	0
$604$	18860.2	1.27055
$605$	2169.18	0.145768
$606$	0	0
$607$	−12884.2	−0.861539	−0.430770	−	0.902462i	$-0.641758\pi$
$607$	−12884.2	−0.861539	−0.430770	+	0.902462i	$0.641758\pi$
$608$	−1949.24	−0.130020
$609$	0	0
$610$	481.595	0.0319659
$611$	−30678.5	−2.03129
$612$	0	0
$613$	−27378.4	−1.80392	−0.901959	−	0.431821i	$-0.857871\pi$
$613$	−27378.4	−1.80392	−0.901959	+	0.431821i	$0.857871\pi$
$614$	1367.62	0.0898906
$615$	0	0
$616$	−225.472	−0.0147476
$617$	24915.7	1.62572	0.812858	−	0.582462i	$-0.197911\pi$
$617$	24915.7	1.62572	0.812858	+	0.582462i	$0.197911\pi$
$618$	0	0
$619$	386.035	0.0250664	0.0125332	−	0.999921i	$-0.496010\pi$
$619$	386.035	0.0250664	0.0125332	+	0.999921i	$0.496010\pi$
$620$	−18950.9	−1.22756
$621$	0	0
$622$	−501.055	−0.0322998
$623$	9683.06	0.622702
$624$	0	0
$625$	−1607.42	−0.102875
$626$	−1148.69	−0.0733402
$627$	0	0
$628$	−10957.7	−0.696271
$629$	−11221.2	−0.711319
$630$	0	0
$631$	−5550.75	−0.350193	−0.175097	−	0.984551i	$-0.556024\pi$
$631$	−5550.75	−0.350193	−0.175097	+	0.984551i	$0.556024\pi$
$632$	3402.85	0.214174
$633$	0	0
$634$	501.121	0.0313912
$635$	44361.3	2.77232
$636$	0	0
$637$	−2607.11	−0.162162
$638$	−294.448	−0.0182717
$639$	0	0
$640$	6662.82	0.411517
$641$	−7892.05	−0.486298	−0.243149	−	0.969989i	$-0.578180\pi$
$641$	−7892.05	−0.486298	−0.243149	+	0.969989i	$0.578180\pi$
$642$	0	0
$643$	14985.7	0.919094	0.459547	−	0.888153i	$-0.348012\pi$
$643$	14985.7	0.919094	0.459547	+	0.888153i	$0.348012\pi$
$644$	−10333.5	−0.632296
$645$	0	0
$646$	−885.723	−0.0539448
$647$	−23441.6	−1.42440	−0.712198	−	0.701979i	$-0.752300\pi$
$647$	−23441.6	−1.42440	−0.712198	+	0.701979i	$0.752300\pi$
$648$	0	0
$649$	−3472.33	−0.210017
$650$	1916.27	0.115634
$651$	0	0
$652$	−19449.5	−1.16825
$653$	−9678.91	−0.580038	−0.290019	−	0.957021i	$-0.593662\pi$
$653$	−9678.91	−0.580038	−0.290019	+	0.957021i	$0.593662\pi$
$654$	0	0
$655$	−5505.63	−0.328432
$656$	−8832.99	−0.525717
$657$	0	0
$658$	740.227	0.0438557
$659$	21735.5	1.28482	0.642408	−	0.766363i	$-0.277936\pi$
$659$	21735.5	1.28482	0.642408	+	0.766363i	$0.277936\pi$
$660$	0	0
$661$	13443.6	0.791066	0.395533	−	0.918452i	$-0.370560\pi$
$661$	13443.6	0.791066	0.395533	+	0.918452i	$0.370560\pi$
$662$	472.971	0.0277682
$663$	0	0
$664$	3813.86	0.222901
$665$	−6985.76	−0.407362
$666$	0	0
$667$	−27046.5	−1.57008
$668$	108.599	0.00629014
$669$	0	0
$670$	−2413.40	−0.139161
$671$	−1611.27	−0.0927012
$672$	0	0
$673$	−1965.43	−0.112573	−0.0562866	−	0.998415i	$-0.517926\pi$
$673$	−1965.43	−0.112573	−0.0562866	+	0.998415i	$0.517926\pi$
$674$	1705.20	0.0974509
$675$	0	0
$676$	−5049.98	−0.287322
$677$	9497.24	0.539156	0.269578	−	0.962979i	$-0.413116\pi$
$677$	9497.24	0.539156	0.269578	+	0.962979i	$0.413116\pi$
$678$	0	0
$679$	−1486.67	−0.0840255
$680$	4554.15	0.256829
$681$	0	0
$682$	−267.701	−0.0150305
$683$	−9717.77	−0.544422	−0.272211	−	0.962238i	$-0.587755\pi$
$683$	−9717.77	−0.544422	−0.272211	+	0.962238i	$0.587755\pi$
$684$	0	0
$685$	−5396.18	−0.300989
$686$	62.9057	0.00350110
$687$	0	0
$688$	−5506.95	−0.305161
$689$	−4898.02	−0.270827
$690$	0	0
$691$	−10090.8	−0.555532	−0.277766	−	0.960649i	$-0.589594\pi$
$691$	−10090.8	−0.555532	−0.277766	+	0.960649i	$0.589594\pi$
$692$	29361.3	1.61293
$693$	0	0
$694$	−1131.05	−0.0618647
$695$	17052.4	0.930697
$696$	0	0
$697$	−12126.3	−0.658991
$698$	−1964.52	−0.106530
$699$	0	0
$700$	10951.0	0.591301
$701$	18004.8	0.970089	0.485044	−	0.874490i	$-0.338803\pi$
$701$	18004.8	0.970089	0.485044	+	0.874490i	$0.338803\pi$
$702$	0	0
$703$	−7200.30	−0.386294
$704$	−5490.42	−0.293932
$705$	0	0
$706$	960.322	0.0511929
$707$	−9285.99	−0.493968
$708$	0	0
$709$	672.259	0.0356096	0.0178048	−	0.999841i	$-0.494332\pi$
$709$	672.259	0.0356096	0.0178048	+	0.999841i	$0.494332\pi$
$710$	−2310.90	−0.122150
$711$	0	0
$712$	−4050.57	−0.213205
$713$	−24589.7	−1.29157
$714$	0	0
$715$	−10492.2	−0.548790
$716$	16272.2	0.849329
$717$	0	0
$718$	−1434.71	−0.0745722
$719$	−9911.49	−0.514098	−0.257049	−	0.966398i	$-0.582750\pi$
$719$	−9911.49	−0.514098	−0.257049	+	0.966398i	$0.582750\pi$
$720$	0	0
$721$	−9522.63	−0.491874
$722$	689.591	0.0355456
$723$	0	0
$724$	35165.9	1.80515
$725$	28662.7	1.46829
$726$	0	0
$727$	21271.5	1.08517	0.542584	−	0.840002i	$-0.317446\pi$
$727$	21271.5	1.08517	0.542584	+	0.840002i	$0.317446\pi$
$728$	1090.59	0.0555221
$729$	0	0
$730$	1188.16	0.0602409
$731$	−7560.18	−0.382522
$732$	0	0
$733$	1013.20	0.0510553	0.0255277	−	0.999674i	$-0.491873\pi$
$733$	1013.20	0.0510553	0.0255277	+	0.999674i	$0.491873\pi$
$734$	1198.31	0.0602595
$735$	0	0
$736$	6488.58	0.324962
$737$	8074.53	0.403567
$738$	0	0
$739$	−30127.0	−1.49965	−0.749825	−	0.661636i	$-0.769862\pi$
$739$	−30127.0	−1.49965	−0.749825	+	0.661636i	$0.769862\pi$
$740$	18472.0	0.917628
$741$	0	0
$742$	118.182	0.00584717
$743$	−26840.0	−1.32525	−0.662627	−	0.748950i	$-0.730558\pi$
$743$	−26840.0	−1.32525	−0.662627	+	0.748950i	$0.730558\pi$
$744$	0	0
$745$	12058.0	0.592983
$746$	1347.69	0.0661424
$747$	0	0
$748$	−7602.37	−0.371618
$749$	−4670.32	−0.227837
$750$	0	0
$751$	−17665.5	−0.858355	−0.429178	−	0.903220i	$-0.641197\pi$
$751$	−17665.5	−0.858355	−0.429178	+	0.903220i	$0.641197\pi$
$752$	36437.3	1.76693
$753$	0	0
$754$	1424.23	0.0687896
$755$	−42441.9	−2.04585
$756$	0	0
$757$	7471.81	0.358742	0.179371	−	0.983782i	$-0.442594\pi$
$757$	7471.81	0.358742	0.179371	+	0.983782i	$0.442594\pi$
$758$	467.556	0.0224042
$759$	0	0
$760$	2922.25	0.139475
$761$	−9239.84	−0.440137	−0.220068	−	0.975484i	$-0.570628\pi$
$761$	−9239.84	−0.440137	−0.220068	+	0.975484i	$0.570628\pi$
$762$	0	0
$763$	−8457.00	−0.401263
$764$	26289.3	1.24491
$765$	0	0
$766$	−553.484	−0.0261073
$767$	16795.5	0.790677
$768$	0	0
$769$	−4109.75	−0.192720	−0.0963599	−	0.995347i	$-0.530720\pi$
$769$	−4109.75	−0.192720	−0.0963599	+	0.995347i	$0.530720\pi$
$770$	253.161	0.0118484
$771$	0	0
$772$	−26833.6	−1.25099
$773$	−30935.4	−1.43942	−0.719709	−	0.694276i	$-0.755725\pi$
$773$	−30935.4	−1.43942	−0.719709	+	0.694276i	$0.755725\pi$
$774$	0	0
$775$	26059.0	1.20783
$776$	621.899	0.0287692
$777$	0	0
$778$	1415.67	0.0652367
$779$	−7781.06	−0.357876
$780$	0	0
$781$	7731.57	0.354235
$782$	2948.38	0.134826
$783$	0	0
$784$	3096.50	0.141058
$785$	24658.5	1.12115
$786$	0	0
$787$	−28615.8	−1.29612	−0.648058	−	0.761591i	$-0.724418\pi$
$787$	−28615.8	−1.29612	−0.648058	+	0.761591i	$0.724418\pi$
$788$	−10894.1	−0.492494
$789$	0	0
$790$	−3820.72	−0.172070
$791$	−6226.76	−0.279896
$792$	0	0
$793$	7793.62	0.349003
$794$	1424.10	0.0636515
$795$	0	0
$796$	36741.0	1.63599
$797$	−14303.2	−0.635690	−0.317845	−	0.948143i	$-0.602959\pi$
$797$	−14303.2	−0.635690	−0.317845	+	0.948143i	$0.602959\pi$
$798$	0	0
$799$	50022.6	2.21486
$800$	−6876.31	−0.303893
$801$	0	0
$802$	2091.87	0.0921029
$803$	−3975.24	−0.174699
$804$	0	0
$805$	23254.0	1.01813
$806$	1294.85	0.0565872
$807$	0	0
$808$	3884.48	0.169128
$809$	−34991.0	−1.52066	−0.760332	−	0.649535i	$-0.774964\pi$
$809$	−34991.0	−1.52066	−0.760332	+	0.649535i	$0.774964\pi$
$810$	0	0
$811$	36535.7	1.58192	0.790962	−	0.611865i	$-0.209580\pi$
$811$	36535.7	1.58192	0.790962	+	0.611865i	$0.209580\pi$
$812$	8139.15	0.351759
$813$	0	0
$814$	260.936	0.0112356
$815$	43768.1	1.88114
$816$	0	0
$817$	−4851.12	−0.207735
$818$	−1504.15	−0.0642925
$819$	0	0
$820$	19961.9	0.850123
$821$	−8067.24	−0.342934	−0.171467	−	0.985190i	$-0.554851\pi$
$821$	−8067.24	−0.342934	−0.171467	+	0.985190i	$0.554851\pi$
$822$	0	0
$823$	−835.480	−0.0353864	−0.0176932	−	0.999843i	$-0.505632\pi$
$823$	−835.480	−0.0353864	−0.0176932	+	0.999843i	$0.505632\pi$
$824$	3983.46	0.168411
$825$	0	0
$826$	−405.250	−0.0170708
$827$	−6645.04	−0.279408	−0.139704	−	0.990193i	$-0.544615\pi$
$827$	−6645.04	−0.279408	−0.139704	+	0.990193i	$0.544615\pi$
$828$	0	0
$829$	35805.8	1.50010	0.750052	−	0.661379i	$-0.230029\pi$
$829$	35805.8	1.50010	0.750052	+	0.661379i	$0.230029\pi$
$830$	−4282.21	−0.179082
$831$	0	0
$832$	26556.8	1.10660
$833$	4251.01	0.176817
$834$	0	0
$835$	−244.385	−0.0101285
$836$	−4878.19	−0.201813
$837$	0	0
$838$	2903.79	0.119701
$839$	−33199.4	−1.36612	−0.683058	−	0.730364i	$-0.739350\pi$
$839$	−33199.4	−1.36612	−0.683058	+	0.730364i	$0.739350\pi$
$840$	0	0
$841$	−3085.98	−0.126532
$842$	2023.06	0.0828018
$843$	0	0
$844$	−38547.2	−1.57210
$845$	11364.2	0.462651
$846$	0	0
$847$	−847.000	−0.0343604
$848$	5817.45	0.235580
$849$	0	0
$850$	−3124.56	−0.126084
$851$	23968.2	0.965476
$852$	0	0
$853$	−10295.8	−0.413274	−0.206637	−	0.978418i	$-0.566252\pi$
$853$	−10295.8	−0.413274	−0.206637	+	0.978418i	$0.566252\pi$
$854$	−188.049	−0.00753500
$855$	0	0
$856$	1953.67	0.0780081
$857$	1707.01	0.0680400	0.0340200	−	0.999421i	$-0.489169\pi$
$857$	1707.01	0.0680400	0.0340200	+	0.999421i	$0.489169\pi$
$858$	0	0
$859$	11934.4	0.474035	0.237017	−	0.971505i	$-0.423830\pi$
$859$	11934.4	0.474035	0.237017	+	0.971505i	$0.423830\pi$
$860$	12445.3	0.493467
$861$	0	0
$862$	2477.90	0.0979089
$863$	−19837.7	−0.782484	−0.391242	−	0.920288i	$-0.627955\pi$
$863$	−19837.7	−0.782484	−0.391242	+	0.920288i	$0.627955\pi$
$864$	0	0
$865$	−66073.0	−2.59717
$866$	−1749.76	−0.0686595
$867$	0	0
$868$	7399.79	0.289361
$869$	12783.0	0.499003
$870$	0	0
$871$	−39056.0	−1.51936
$872$	3537.69	0.137387
$873$	0	0
$874$	1891.88	0.0732194
$875$	−8957.43	−0.346076
$876$	0	0
$877$	5630.62	0.216799	0.108399	−	0.994107i	$-0.465427\pi$
$877$	5630.62	0.216799	0.108399	+	0.994107i	$0.465427\pi$
$878$	−3023.72	−0.116225
$879$	0	0
$880$	12461.7	0.477368
$881$	−42058.5	−1.60838	−0.804191	−	0.594370i	$-0.797401\pi$
$881$	−42058.5	−1.60838	−0.804191	+	0.594370i	$0.797401\pi$
$882$	0	0
$883$	15501.7	0.590797	0.295398	−	0.955374i	$-0.404548\pi$
$883$	15501.7	0.590797	0.295398	+	0.955374i	$0.404548\pi$
$884$	36772.2	1.39907
$885$	0	0
$886$	−2880.96	−0.109241
$887$	−27103.2	−1.02597	−0.512986	−	0.858397i	$-0.671461\pi$
$887$	−27103.2	−1.02597	−0.512986	+	0.858397i	$0.671461\pi$
$888$	0	0
$889$	−17321.8	−0.653492
$890$	4547.99	0.171291
$891$	0	0
$892$	−14294.3	−0.536557
$893$	32097.9	1.20282
$894$	0	0
$895$	−36618.0	−1.36760
$896$	−2601.64	−0.0970029
$897$	0	0
$898$	1806.59	0.0671346
$899$	19367.9	0.718526
$900$	0	0
$901$	7986.45	0.295302
$902$	281.982	0.0104091
$903$	0	0
$904$	2604.75	0.0958327
$905$	−79135.3	−2.90668
$906$	0	0
$907$	43553.1	1.59444	0.797219	−	0.603690i	$-0.206304\pi$
$907$	43553.1	1.59444	0.797219	+	0.603690i	$0.206304\pi$
$908$	−10332.8	−0.377649
$909$	0	0
$910$	−1224.52	−0.0446072
$911$	44312.9	1.61158	0.805792	−	0.592199i	$-0.201740\pi$
$911$	44312.9	1.61158	0.805792	+	0.592199i	$0.201740\pi$
$912$	0	0
$913$	14327.0	0.519337
$914$	−962.458	−0.0348307
$915$	0	0
$916$	−20952.5	−0.755777
$917$	2149.79	0.0774179
$918$	0	0
$919$	−2487.86	−0.0893004	−0.0446502	−	0.999003i	$-0.514217\pi$
$919$	−2487.86	−0.0893004	−0.0446502	+	0.999003i	$0.514217\pi$
$920$	−9727.53	−0.348595
$921$	0	0
$922$	964.428	0.0344487
$923$	−37397.1	−1.33363
$924$	0	0
$925$	−25400.5	−0.902878
$926$	−1616.82	−0.0573780
$927$	0	0
$928$	−5110.68	−0.180783
$929$	−53068.5	−1.87419	−0.937093	−	0.349079i	$-0.886495\pi$
$929$	−53068.5	−1.87419	−0.937093	+	0.349079i	$0.886495\pi$
$930$	0	0
$931$	2727.73	0.0960235
$932$	29534.2	1.03801
$933$	0	0
$934$	476.767	0.0167027
$935$	17107.9	0.598385
$936$	0	0
$937$	44976.4	1.56810	0.784052	−	0.620695i	$-0.213149\pi$
$937$	44976.4	1.56810	0.784052	+	0.620695i	$0.213149\pi$
$938$	942.364	0.0328031
$939$	0	0
$940$	−82345.6	−2.85725
$941$	−20420.9	−0.707442	−0.353721	−	0.935351i	$-0.615084\pi$
$941$	−20420.9	−0.707442	−0.353721	+	0.935351i	$0.615084\pi$
$942$	0	0
$943$	25901.4	0.894451
$944$	−19948.2	−0.687774
$945$	0	0
$946$	175.802	0.00604211
$947$	−2792.83	−0.0958340	−0.0479170	−	0.998851i	$-0.515258\pi$
$947$	−2792.83	−0.0958340	−0.0479170	+	0.998851i	$0.515258\pi$
$948$	0	0
$949$	19228.0	0.657709
$950$	−2004.93	−0.0684721
$951$	0	0
$952$	−1778.26	−0.0605397
$953$	37731.4	1.28252	0.641260	−	0.767324i	$-0.278412\pi$
$953$	37731.4	1.28252	0.641260	+	0.767324i	$0.278412\pi$
$954$	0	0
$955$	−59159.9	−2.00458
$956$	22661.1	0.766645
$957$	0	0
$958$	−1158.47	−0.0390693
$959$	2107.05	0.0709491
$960$	0	0
$961$	−12182.5	−0.408932
$962$	−1262.13	−0.0423001
$963$	0	0
$964$	−14169.8	−0.473421
$965$	60384.9	2.01436
$966$	0	0
$967$	21417.8	0.712254	0.356127	−	0.934438i	$-0.384097\pi$
$967$	21417.8	0.712254	0.356127	+	0.934438i	$0.384097\pi$
$968$	354.313	0.0117645
$969$	0	0
$970$	−698.269	−0.0231135
$971$	775.461	0.0256290	0.0128145	−	0.999918i	$-0.495921\pi$
$971$	775.461	0.0256290	0.0128145	+	0.999918i	$0.495921\pi$
$972$	0	0
$973$	−6658.47	−0.219384
$974$	1222.26	0.0402090
$975$	0	0
$976$	−9256.60	−0.303582
$977$	23964.4	0.784739	0.392369	−	0.919808i	$-0.371655\pi$
$977$	23964.4	0.784739	0.392369	+	0.919808i	$0.371655\pi$
$978$	0	0
$979$	−15216.2	−0.496744
$980$	−6997.87	−0.228101
$981$	0	0
$982$	1175.81	0.0382092
$983$	−11537.1	−0.374339	−0.187169	−	0.982328i	$-0.559931\pi$
$983$	−11537.1	−0.374339	−0.187169	+	0.982328i	$0.559931\pi$
$984$	0	0
$985$	24515.4	0.793022
$986$	−2322.27	−0.0750062
$987$	0	0
$988$	23595.5	0.759791
$989$	16148.3	0.519198
$990$	0	0
$991$	−13148.8	−0.421479	−0.210740	−	0.977542i	$-0.567587\pi$
$991$	−13148.8	−0.421479	−0.210740	+	0.977542i	$0.567587\pi$
$992$	−4646.43	−0.148714
$993$	0	0
$994$	902.338	0.0287932
$995$	−82679.9	−2.63430
$996$	0	0
$997$	−12337.2	−0.391897	−0.195949	−	0.980614i	$-0.562779\pi$
$997$	−12337.2	−0.391897	−0.195949	+	0.980614i	$0.562779\pi$
$998$	1257.35	0.0398806
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.q.1.3		5
3.2	odd	2		231.4.a.j.1.3	✓	5
21.20	even	2		1617.4.a.o.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.j.1.3	✓	5	3.2	odd	2
693.4.a.q.1.3		5	1.1	even	1	trivial
1617.4.a.o.1.3		5	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-0.183399 q^{2} -7.96636 q^{4} +17.9271 q^{5} -7.00000 q^{7} +2.92821 q^{8} -3.28780 q^{10} +11.0000 q^{11} -53.2063 q^{13} +1.28379 q^{14} +63.1939 q^{16} +86.7553 q^{17} +55.6681 q^{19} -142.814 q^{20} -2.01738 q^{22} -185.307 q^{23} +196.380 q^{25} +9.75796 q^{26} +55.7646 q^{28} +145.956 q^{29} +132.697 q^{31} -35.0153 q^{32} -15.9108 q^{34} -125.490 q^{35} -129.344 q^{37} -10.2094 q^{38} +52.4942 q^{40} -139.776 q^{41} -87.1438 q^{43} -87.6300 q^{44} +33.9850 q^{46} +576.595 q^{47} +49.0000 q^{49} -36.0158 q^{50} +423.861 q^{52} +92.0572 q^{53} +197.198 q^{55} -20.4975 q^{56} -26.7680 q^{58} -315.667 q^{59} -146.479 q^{61} -24.3365 q^{62} -499.129 q^{64} -953.834 q^{65} +734.048 q^{67} -691.124 q^{68} +23.0146 q^{70} +702.870 q^{71} -361.385 q^{73} +23.7214 q^{74} -443.472 q^{76} -77.0000 q^{77} +1162.09 q^{79} +1132.88 q^{80} +25.6347 q^{82} +1302.46 q^{83} +1555.27 q^{85} +15.9820 q^{86} +32.2103 q^{88} -1383.29 q^{89} +372.444 q^{91} +1476.22 q^{92} -105.747 q^{94} +997.965 q^{95} +212.382 q^{97} -8.98653 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26}+ \cdots + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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